We model both the ballistic and dynamic versions of the BoE task with independent Reinforcement Learning (RL) agents having access only to the precedent episode outcome (high reward, low reward, tie) and to the \(y\)  position of all agents in the dynamic case. In this sense, we are modelling also the dynamic changes of decisions something that we did not address in a precedent paper [1]}.
For the RL algorithm we use Q learning, which is an off-policy model-free algorithm with an epsilon greedy policy. Q-learning maintains a function (in our case there will be two functions, one for each agent), \(Q(s,a)\) , that given a state and an action, returns the future discounted reward that one can obtain from applying that action in that state. The Q function is updated using the following equation derived from the bellman equations:
\(Q(s,a) \leftarrow Q(s,a) + \mu (r^i+\gamma \max _{a^{\prime }} Q(s^{\prime },a^{\prime })-Q(s,a))\) 
